Remarks on Kreĭn's inequality (Q2429453)

Let \(\cos\phi_{xy}= \text{Re}\langle x,y\rangle/\| x\|\cdot\| y\|\) and \(\cos\psi_{xy}= |\langle x,y\rangle|/\| x\|\cdot\| y\|\) for \(x,y\in \mathbb{C}^n\setminus\{0\}\). \textit{M. G. Kreĭn's} inequality [Funct. Anal. Appl. 3, 73-74 (1969); translation from Funkts. Anal. Prilozh. 3, No. 1, 89--90 (1969; Zbl 0182.17402)] states that \(\phi_{xy}\leq \phi_{xz}+ \phi_{yz}\) for any \(x,y,z\neq 0\). The author remarks that, a similar result holds true for \(\psi_{xy}\), and gives an application of these two inequalities. This is in fact the triangle inequality for the corresponding chordal metrics on the unit ball of \(\mathbb{C}^n\). A generalization is also pointed out.